document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1145662>Question 1145662</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Aaron and Jonathan take turns shooting a free throw, with Aaron going first. Whoever makes the first shot wins. The game continues until<BR>\n" );
document.write( "someone makes the shot. Aaron has a 40% chance of making a free throw, while Jonathan has a 50% chance. What is Aaron’s chance of<BR>\n" );
document.write( "winning the game?  </SPAN>\n" );
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document.write( "P(Aaron wins on his first shot) = 40% = 2/5<br><BR>\n" );
document.write( "P(Aaron wins on his second shot) = (3/5)*(1/2)*(2/5) = 3/25<BR>\n" );
document.write( "(Aaron misses on his first, probability 3/5; then Jonathan misses on his first, probability 1/2; then Aaron makes his second shot probability 2/5)<br><BR>\n" );
document.write( "P(Aaron wins on his third shot) = (3/5)*(1/2)*(3/5)*(1/2)*(2/5) = 9/250<br><BR>\n" );
document.write( "etc., etc., etc.....<br><BR>\n" );
document.write( "The sum of the probabilities is a geometric progression with first term 2/5 and common ratio (3/5)*(1/2) = 3/10.<br><BR>\n" );
document.write( "The probability that Aaron wins is the sum of that infinite geometric progression:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2F5%29%2F%281-%283%2F10%29%29+=+%284%2F10%29%2F%287%2F10%29+=+4%2F7 ALIGN=MIDDLE  ALT=\"%282%2F5%29%2F%281-%283%2F10%29%29+=+%284%2F10%29%2F%287%2F10%29+=+4%2F7\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );